Harmonic Coordinates - Pixar Graphics Technologies

Harmonic Coordinates
Tony DeRose Mark Meyer
Pixar Technical Memo #06-02
Pixar Animation Studios
(a) (b) (c) (d)
Figure 1: A character (shown in blue) being deformed by a cage (shown in black) using harmonic coordinates. (a) The character and cage at
bind-time; (b) - (d) the deformed character corresponding to three different poses of the cage.
Abstract
Generalizations of barycentric coordinates in two and higher dimensions
have been shown to have a number of applications in
recent years, including finite element analysis, the definition of Spatches
(n-sided generalizations of Bézier surfaces), free-form deformations,
mesh parametrization, and interpolation. In this paper
we present a new form of d dimensional generalized barycentric coordinates.
The new coordinates are defined as solutions to Laplace’s
equation subject to carefully chosen boundary conditions. Since solutions
to Laplace’s equation are called harmonic functions, we call
the new construction harmonic coordinates. We show that harmonic
coordinates possess several properties that make them more attractive
than mean value coordinates when used to define two and three
dimensional deformations.
Keywords: Barycentric coordinates, mean value coordinates,free
form deformations, rigging.
1 Introduction
Two dimensional barycentric coordinates are fundamental in a wide
variety of applications, including Gouraud shading of triangles and
the definition of triangular Bézier patches [Farin 2002]. Given a
triangle with vertices T1,T2,T3, barycentric coordinates allow every
point p in the plane of the triangle to be expressed uniquely as
p = ∑ βi(p)Ti
(1)
i=1,2,3
where the numbers β1(p),β2(p),β3(p) are the barycentric coordinates
of p with respect to T1,T2,T3. They can be defined in many
ways, one of the simplest being as the unique linear functions satisfying
the interpolation conditions:
βi(Tj) = δi, j, i, j = 1,2,3. (2)
Similarly, barycentric coordinates in three dimensions can be
defined relative to a non-degenerate tetrahedron with vertices
T1,T2,T3,T4 ∈ ℜ 3 as the unique linear functions that satisfy Equation
2 where the indices i and j run from 1 to 4 instead of from 1 to
3.
As described in Ju et. al. [Ju et al. 2005], most of the uses of
barycentric coordinates stem from their use in the construction of
interpolating functions. Gouraud shading is a familiar example
where colors c1,c2,c3 assigned to the vertices of a triangle are interpolated
across the triangle according to
c(p) = ∑
i=1,2,3
βi(p)ci
(3)
A similar formula can be used to define a deformation of two-space.
Namely, let T ′
1 ,T ′
2 and T ′
3 denote new positions for the vertices of
the original triangle. The deformed position of point p can then be
defined as
p ′ = ∑
i=1,2,3
βi(p)T ′
i
(4)
Consider now the problem of defining such two dimensional coordinates
(and corresponding interpolants) relative to polygons with
more than three vertices. The interpolation conditions are still important,
but it is no longer sufficient to require the coordinate functions
to be linear – there are too many interpolation conditions to
satisfy with linear functions. An interesting and important question
then is how to generalize barycentric coordinates to arbitrary closed
polygons in the plane and arbitrary closed polyhedra in space.
The problem of generalizing barycentric coordinates is surprisingly
rich, and has received considerable attention in recent years
([Wachpress 1975], [R.Sibson 1981], [Loop and DeRose 1989],
[Warren 1996], [Meyer et al. 2002], [Floater 2003], [Floater et al.
2005], [Ju et al. 2005]). Ju et. al. provide a good overview in [Ju
et al. 2005]. Since not all properties of barycentric coordinates can

(a) (d)
(b) (e)
(c) (f)
Figure 2: Two dimensional generalized barycentric coordinates
used to define deformations of two different objects (shown in blue)
using cages (shown in black). The top row shows the cages and objects
at ”bind” time. The second row shows modified cages and
the corresponding deformed objects using mean value coordinates.
The last row shows modified cages and deformed objects using harmonic
coordinates. In the last two rows, the original undeformed
object is shown in white. The two methods perform similarly for
convex shapes. In the bipedal case, harmonic coordinates perform
better in that the motion of cage points in the left leg does not influence
points in the right leg.
be retained in the generalization, the richness results from the many
different ways the properties can be relaxed.
We are particularly interested in using generalized barycentric coordinates
for character deformation, as shown in Figure 1 and as
described in Ju et. al [Ju et al. 2005]. In this application, an object
to be deformed is positioned relative to a closed shape that we’ll call
a cage. Examples are shown in Figures 1 and Figure 2. The object
is then “bound” to the cage by computing generalized barycentric
coordinates gi(p) of each object point p relative to the cage vertices
Ci. As the cage vertices are moved to new locations C ′ i , the
deformed points p ′ are computed from
p ′ = ∑ i
gi(p)C ′ i
Of the various generalized barycentric coordinate formulations
available, mean value coordinates [Floater 2003; Floater et al. 2005;
Ju et al. 2005] are particularly useful in this application because:
(5)
• The cage that controls the deformation can be any simple
closed polygon in two dimensions, and any simple closed triangular
mesh in three dimensions.
• The coordinates are smooth, so the deformation is smooth.
• The coordinates reproduce linear functions, so the object
doesn’t “pop” when it is bound. That is, the coordinates are
such that setting C ′ i to Ci in Equation 5 results in p ′ reducing
to p.
A second example motivated by the articulation of bipedal characters
is shown in the second column of Figure 2. Notice how the
modified cage points on the leg on the left in Figure 2(e) influence
the position of object points in the leg on the right. This occurs
because mean value coordinates are based on Euclidean (straightline)
distances between points of the cage and points of the object.
Although the influence is noticeable in still images, the movement
of points in the right leg when the left leg cage points is particularly
striking in interactive use, as demonstrated in the accompanying
video. The behavior of 3D mean value coordinates is similar, and is
highly undesirable for the articulation of characters in feature film
production.
What is needed for character articulation is a form of generalized
barycentric coordinates that adds the following properties to those
enjoyed by mean value coordinates:
• Non-negativity. This implies that object points move in the
same direction as cage points; negative coordinates mean
they can move in the opposite direction. Negativity of mean
value coordinates is responsible for the right leg points in Figure
2(e) moving in the opposite direction from the left leg cage
points. Negativity is also responsible for the collapsing of the
left leg points in Figure 2(e).
• Interior locality. Informally, the coordinates should fall off
as a function of the distance between cage points and object
points as measured within the cage.
In this paper we show that such coordinates can be produced as solutions
to Laplace’s equation with appropriately chosen boundary
conditions. Since solutions to Laplace’s equation are generically
referred to as harmonic functions, we therefore call these coordinates
harmonic coordinates, and the deformations they generate
harmonic deformations. 1
1.1 Previous work
Laplace’s equation, harmonic functions, and harmonic maps have
often been mentioned in previous constructions of barycentric coordinates
in two dimensions. For instance, the “cotangent weights” of
[Pinkhall and Polthier 1993] and [Meyer et al. 2002] can be derived
from piecewise linear discretizations of Laplace’s equation. Similarly,
Floater’s construction of mean value coordinates was motivated
by the mean value theorem for harmonic functions. It is
somewhat surprising to us that direct solution of Laplace’s equation
has never been used to create generalized barycentric coordinates,
but that seems to be the case.
Another connection between Laplace’s equation and mean value
coordinates comes from the motivation given in [Ju et al. 2005].
They derive mean value coordinates starting with an interpolant
they call the mean value interpolant. The mean value interpolant
to a function f defined on a closed boundary works as follows. To
1 Since each component of a harmonic deformation is a harmonic function,
many texts refer to such deformations as harmonic maps. We prefer
the term harmonic deformation because of the context in which they’re used
in this paper.

p
(a)
x
Figure 3: Mean value vs harmonic interpolation. (a) The straightline
paths corresponding to mean value interpolation. (b) The
Brownian paths corresponding to harmonic interpolation.
compute a value for each interior point p, consider each point x on
the boundary. Multiply f (x) by the reciprocal distance from x to
p, then average over all x (see Figure 3(a)). This definition makes
it clear that mean value coordinates involve straight-line distances
irrespective of the visibility of x from p. An alternative interpolant
that respects visibility is to average not over all straight-line paths,
but rather to average over all Brownian paths leaving p, where the
value assigned to each path is the value of f at the point the path first
hits the boundary (see Figure 3(b)). Although this definition at first
seems intractable to compute, it is a famous result from stochastic
processes (c.f. [Port and Stone 1978], [Bass 1995]) that the interpolant
thus produced (in any dimension) in fact satisfies Laplace’s
equation subject to the boundary conditions given by f . 2
2 Theory
In this section we formalize the discussion of Section 1. Let C be
a closed (not necessarily convex) volume in d dimensions with a
piecewise linear boundary. Geometers call such shapes polytopes,
but because we have specific uses in mind, we refer to these shapes
instead as cages. In two dimensions, a cage is a region of the plane
bounded by a closed polygon (such as the one shown in Figure 2),
and in three dimensions a cage is a closed region of space bounded
by planar (though not necessarily triangular) faces. For each of
the vertices Ci of the cage, we seek a function hi(p) defined on C
subject to the following conditions:
1. Interpolation: hi(Cj) = δi, j.
2. Affine-invariance: ∑i hi(p) = 1 for all p ∈ C.
3. Strict generalization of barycentric coordinates: when C is a
simplex, hi(p) is the barycentric coordinate of p with respect
to Ci.
4. Smoothness: The functions hi(p) are at least C 1 smooth.
5. Non-negativity: hi(p) ≥ 0, for all p ∈ C.
6. Linear reproduction: Given an arbitrary function f (p), the
coordinate functions can be used to define an interpolant
H[ f ](p) according to:
H[ f ](p) = ∑hi(p) f (Ci) (6)
i
Following Ju et. al [Ju et al. 2005], we require H[ f ](p) to
be exact for linear functions. As shown by Ju et. al, taking
f (p) = p means that
p = ∑hi(p)Ci (7)
i
which is the“non-popping” condition mentioned in Section 1.
2 We thank [name omitted for review purposes] for pointing out this con-
nection to us.
p
(b)
7. Interior locality: We quantify the notion of interior locality
introduced above as follows: interior locality holds, if, in addition
to non-negativity, the coordinate functions have no interior
extrema.
Mean value coordinates possess all but two of these properties:
namely, non-negativity and interior locality. We claim that coordinate
functions satisfying all seven properties can be obtained as
solutions to Laplace’s equation
▽ 2 hi(p) = 0, p ∈ Int(C) (8)
if the boundary conditions are carefully chosen.
To gain some insight into how the boundary conditions are determined,
we consider first the construction of harmonic coordinates
in two dimensions. It will then be clear how the construction generalizes
to d dimensions. For reasons that will soon become apparent,
the appropriate boundary conditions for hi(p) in two dimensions are
as follows. Let ∂ p denote a point on the boundary ∂C of C, then
hi(∂ p) = φi(∂ p),for all∂ p ∈ ∂C (9)
where φi(∂ p) is the (univariate) piecewise linear function such that
φi(Cj) = δi, j. For example, if C is the cage shown in Figure 4(a),
then φi(∂ p) is the piecewise linear function defined on the edges
e1,...,e15 such that φi(Cj) = δi, j, for i, j = 1,...,15.
We now show that functions satisfying Equation 8 subject to Equation
9 possess the properties enumerated above. It turns out that the
linear reproduction property subsumes several other conditions, so
for purposes of proof we verify the conditions in a different order
than the one presented above.
• Interpolation: by construction hi(Cj) = φi(Cj) = δi, j.
• Smoothness: Away from the boundary harmonic coordinates
are solutions to Laplace’s equation, and hence they are C ∞ .
• Non-negativity: harmonic functions achieve their extrema at
their boundaries. Since boundary values are restricted to [0,1],
interior values are also restricted to [0,1].
• Linear reproduction: Let f (p) be an arbitrary linear function.
We need to show that H[ f ](p) = f (p), where H[ f ](p)
is defined as in Equation 6. We begin by establishing that
H[ f ](p) = f (p) everywhere on the boundary of C. If ∂ p is a
point on the boundary of C, then by construction
H[ f ](∂ p) = ∑hi(∂ p) f (Ci) = ∑φi(∂ p) f (Ci) (10)
i
i
The functions φi(∂ p) are the univariate linear B-spline basis
functions (commonly known as the “hat function” basis),
which are capable of reproducing all linear functions on ∂C
(in fact, they reproduce all piecewise linear functions on ∂C).
Next we extend the result to the interior of C. Note that since
f (p) is linear, all second derivatives vanish, and in particular
▽ 2 f (p) = 0; thus f (p) satisfies Laplace’s equation on the
interior of C. H[ f ](p) also satisfies Laplace’s equation on the
interior, because for interior points ·p:
▽ 2 H[ f ](·p) = ▽ 2 ∑ i
= ∑ i
= ∑ f (Ci)0
i
= 0
hi(·p) f (Ci)
f (Ci) ▽ 2 hi(·p)
Since f (p) and H[ f ](p) agree on their boundaries and are
both solutions to the same differential equation, by uniqueness
of solutions to PDEs, they must be the same function.

e15
C1
C15
e2
e1
C3
C2
(a) (b) (c)
Figure 4: A comparison of coordinate functions for a concave cage. (a) A 2D cage with vertices C1,...,C15; (b) the value of the mean
value coordinate for C2 (yellow indicates positive values, green indicates negative values); (c) the value of the harmonic coordinate for C2
(red denotes the exterior of the cage where the function is undefined). To accentuate values near zero, intensities of yellow and green are
proportional to the square root of the coordinate function value. The significant influence of the position of C2 on object points in the leg on
the right is indicated by the presence of green in the right leg of (b). The corresponding influence in (c) is essentially zero.
• Affine invariance: The function f (p) = 1 is linear, so affine
invariance follows immediately from the linear reproduction
property.
• Strict generalization of barycentric coordinates. If the cage
C consists of a single triangle harmonic coordinates reduce
to barycentric coordinates. Let β j(p) denote the barycentric
coordinates of p with respect to the triangle. To establish that
h j(p) = β j(p), note that β j(p) is a linear function, so we can
use the linear reproduction property above by taking f (p) =
β j(p):
β j(p) = H[β j](p)
= ∑ i
hi(p)β j(Ci)
= ∑hi(p)δi, j
i
= h j(p)
• Interior locality: follows from non-negativity and the fact that
harmonic functions possess no interior extrema.
To generalize from two to d dimensions, we first back up and consider
harmonic coordinates in one dimension. In one dimension
a cage is a line segment bounded by two vertices C0 and C1, and
Laplace’s equation reduces to
d2hi(p) = 0. (11)
d p2 Thus, hi(p) is a linear function, and the proper (zero dimensional)
boundary conditions come from the interpolation property:
hi(Cj) = δi, j.
With this insight, we can repose the two dimensional construction
as: to construct two dimensional harmonic coordinates, start with
the interpolation conditions hi(Cj) = δi, j. This determines the coordinates
on the 0-dimensional facets (the vertices) of C. Next, extend
the coordinates to the 1-dimensional facets (the edges) of C using
the one dimensional version of Laplace’s equation. Finally, extend
them to the two dimensional facets (the interior) of C using the two
dimensional version of Laplace’s equation.
The extension to three and higher dimensions follows immediately:
The harmonic coordinates hi(p) for a d dimensional cage C with
vertices Ci, are the unique functions such that:
1. hi(Cj) = δi, j.
2. On every facet of dimension k ≤ d, the k dimensional Laplace
equation is satisfied.
To prove that d dimensional harmonic coordinates defined in this
way possess the required properties, we can use induction on the
facet dimension, starting with the 1-facets as the base case. The
proofs given above for two dimensions are actually more general;
they are valid in any dimension k assuming that linear reproduction
is achieved on the k − 1 facets. These proofs therefore serve as the
inductive step.
Having defined coordinates in this way, by construction we have
the following additional property that is shared by barycentric coordinates
and Warren’s [Warren 1996] construction:
• Dimension reduction: d dimensional harmonic coordinates,
when restricted to a k < d dimensional facet, reduce to k dimensional
harmonic coordinates.
For example, a three dimensional cage bounded by triangular facets
possesses harmonic coordinates that reduce to barycentric coordinates
on the faces. Similarly, a dodecahedral cage will have 3D
harmonic coordinates that reduce to 2D harmonic coordinates on
its pentagonal faces.
3 Implementation
Our current implementation of harmonic coordinates is limited to
two and three dimensions. In both cases we use a simple hierarchical
finite difference solver, though in principle any solution method
for Laplace’s equation, such as a finite element method, could be
used.
Now for some details. First we’ll describe the non-hierarchial version
of the solver. We’ll then describe the extension to the hierarchical
solver. For each vertex Ci of the cage, we approximate hi(p)
over the interior of the cage as follows:
1. Allocate a regular grid of cells that is large enough to enclose
the cage. We choose the grid to contain 2 s cells on a side. All
two dimensional examples have been computed with s = 6;
three dimensional examples use s = 7. Each grid cell contains
a value, and a tag, where the tag is one of UNTYPED,
BOUNDARY, INTERIOR, or EXTERIOR.
2. Initialize the grid by:
(a) Tag all cells as UNTYPED.

(b) Scan-convert boundary conditions into the grid, marking
each scan converted cell with the BOUNDARY tag.
In two dimensions, the function φi(p) as defined in Section
2 is scan-converted into the grid. In three dimensionals,
our implementation is currently restricted to triangular
faces, meaning that the boundary values varying
in a piecewise linear fashion. We therefore use a
simple voxel-based triangle scan-converter in this stage.
(c) Starting with one of the corner cells, flood fill the exterior,
marking each visited cell with the EXTERIOR tag.
The flood fill recursion stops when BOUNARY tags are
reached. Since the boundary is closed, only the exterior
cells are visited during this stage.
(d) Mark remaining UNTYPED cells as INTERIOR with
harmonic coordinate value equal to 0.
3. Laplacian smooth: For each INTERIOR cell, replace the
value of the cell with the average of the value of its neighbors.
In 2D cells are considered to be 4-connected; in 3D they
are considered to be 6-connected. This Laplacian smoothing
step is performed iteratively until the termination criterion is
reached. Our solver terminates when the average change to a
cell drops below a specified threshold τ. All examples in this
paper have used τ = 10 −5 .
The solver described above can be significantly accelerated by noting
that Laplace’s equation produces very smoothly varying functions.
By first solving the problem at a lower resolution, better starting
points for the iteration can be obtained. The hierarchical solver
exploits this observation by “pulling” the boundary conditions up to
a coarser level, recursively solving there, “pushing” the coarse solution
down to the finer level, then iterating the Laplacian smoothing
step until convergence is reached.
The pulling step in two dimensions computes a coarse level grid of
size 2 s−1 × 2 s−1 from a fine level grid of size 2 s × 2 s . Each coarse
level grid cell represents four “children” cells on the finer level. In
three dimensions, each coarse level grid cell represents eight children
cells on the finer level. In both cases, a coarse cell is tagged
as a BOUNDARY if at least one child is tagged as a BOUNDARY;
it is tagged as EXTERIOR if all children are EXTERIOR, and it
is tagged as INTERIOR if all children are INTERIOR. The value
of a coarse level BOUNDARY cell is the average of the finer level
BOUNDARY cells. INTERIOR cells on the coarse level are initialized
with a value of zero.
The pushing step propagates values from coarse level cells to IN-
TERIOR cells on the finer level. Specifically, all INTERIOR cells
on the finer level receive the value of their parent cell on the coarse
level.
4 Results
The behavior of harmonic deformations in two dimensions is illustrated
in the accompanying video as well as in Figure 2. The behavior
of three dimensional harmonic deformations is illustrated in
Figure 1, where we have bound an object containing 8019 vertices
to a cage containing 112 vertices. The binding time for this example
was 262 seconds, using the hierarchical solver with a finest
grid with 2 7 cells on a side, and a coarsest grid with 2 4 cells on
a side. The termination tolerance τ was 10 −5 . The corresponding
bind time for mean value coordinates was 443 seconds using the
algorithm as published in Figure 4 of [Ju et al. 2005].
Notice that the bind time for harmonic coordinates is faster than
that for mean value coordinates in this case. The primary reason is
that the harmonic coordinate solver computes an entire coordinate
Subdivisions Object vertices MVC (in sec) HC (in sec)
0 21 0.16 29
2 242 1.7 30
4 3842 24 30
5 15,362 113 30
Table 1: A comparison of the binding time of the mean value
and harmonic coordinate solvers as the number of object points increases.
These examples were generated by using an icosahedron as
the cage, and a subdivided dodecahedron as the object. The “Subdivisions”
column indicates the number of times the dodecahedron
was subdivided. Note that the time required for harmonic coordinates
is relatively insensitive to the number of object vertices.
function at a time. Once a coordinate function is computed it is
very inexpensive to look up the value for each of the object points.
The running time of the harmonic solver is therefore most strongly
dependent on the number of cage vertices. The mean value solver,
on the other hand, iterates over the entire cage for each of the object
points. When the number of object points is small compared to
the number of cage points, the mean value solver is faster. As the
number of object points increases, the harmonic solver eventually
outperforms the mean value solver. This trend is demonstrated in
Table 1.
One potential disadvantage of harmonic coordinates compared to
mean value coordinates is memory overhead. A straightforward
implementation of mean value coordinates requires only a constant
amount of additional memory for each face of the cage, whereas the
memory requirements of our simple harmonic solver is dominated
by the solver grid. In two dimensions the grids are typically small
(roughly 40Kbytes in our examples), but in three dimensions the
grids can become rather large (roughly 25Mbytes in our examples).
Harmonic coordinates computed as above are only numerical approximations,
where the cell size and termination threshold determine
the accuracy. The approximation error will, in general, cause
each object point p to experience a residual ∆(p) when it is bound
to the cage:
∆(p) = p −∑ hi(p)Ci
(12)
i
Another source of error occurs when coordinates below a threshold
are removed from the sum, a process we call sparsification. Residuals
due to sparsification occur for both mean value coordinates and
harmonic coordinates. In cases where the residuals are too large,
either because of an inaccurate solve or because of overly agressive
sparisification, the residuals can be computed and stored at bind
time on a per object point basis. They can then be added back at
deformation time to improve the accuracy of the deformation with
little run-time overhead. The examples used in this paper and the
accompanying video were accurate enough that residuals were not
used.
4.1 Extension to cell complexes
Harmonic coordinates as formulated thus far are defined relative to
a cage consisting of a polytope, meaning the deformation is controlled
entirely by boundary vertices of the cage. Once the behavior
of the boundary is set, the behavior on the entire interior is completely
determined. In many instances this is ideal. However, it
is sometimes helpful to give artists additional control over interior
details of the deformation. A simple example is shown in Figure 5
where an additional isolated vertex has been added to refine control
of the deformation in the area of the head of the character.
It is also possible to extend the cage to include interior faces and
edges; an example of including a collection of interior edges is

(a) (b)
Figure 5: An example of interior control. (a) shows the cage and
object at bind time, where an isolated interior vertex has been added
to the cage; (b) shows the deformed object in response to movement
of the interior vertex.
shown in Figure 6. As demonstrated in this figure, the interior controls
need not form a manifold — it is sufficient for the interior
of the cage to form what is known as a linear cell complex. Intuitively,
a linear cell complex is a collection of “cells” (vertices, linear
edges, and planar faces) with the property that the intersection
of any two cells is either empty or is another cell in the collection.
Harmonic coordinates are easily adapted to such cages by treating
the interior facets in exactly the same way as the bounary. 3
Since harmonic functions are guaranteed to be only continuous at
interior boundary conditions, harmonic coordinates are only C 0
smooth across interior facets. In practice this means that if interior
facets are used and smoothly deformed objects are desired, the
interior facets should be placed so the object being deformed does
not cross them.
5 Summary
We have provided a new and easy to implement construction for
generalized barycentric coordinates as solutions to Laplace’s equation
subject to carefully chosen boundary conditions — ones that
correspond to lower dimensional solutions to Laplace’s equation.
These harmonic coordinates improve on mean value coordinates in
that they are guaranteed to be positive everywhere in the interior
of the cage, and their influence falls off with distance as measured
within the cage. Moreover, we have shown that the construction
of harmonic coordinates can be carried out in any dimension, and
we’ve show that the cage can be augmented with additional interior
vertices, edges, and faces to provide more detailed control when
necessary.
Unlike mean value coordinates, harmonic coordinates are defined
only within the cage, and they do not possess a closed form expression.
The memory requirements of harmonic coordinates are also
considerably larger than mean value coordinates, especially in three
dimensions. However, we’ve show that they can be efficiently approximated
using a hierarchical solver, and we’ve shown that in the
common case where the number of object points greatly exceeds
the number of cage points, they are faster to compute than mean
value coordinates. Once coordinates have been computed, the cost
of evaluating harmonic deformations is identical to that of deformations
based on mean value coordinates.
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3 Note to reviewers: we have not yet implemented harmonic coordinates
for cell complexes in 3D, but we will have such examples prior to final
publication.
(a)
(b) (c)
Figure 6: An example of interior control using a linear cell complex.
(a) An object deformed using a cage with no interior controls.
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